ST2120 Assume X1, X2, …, X81 to be 81 independent random samples selected from a population with a mean of 20: Data science for business analytics II Assignment, UOG, Ireland

Question 1 – Compulsory

IN EACH OF QUESTIONS 1 TO 10 BELOW, WRITE DOWN ONE CHOICE OF ANSWER. FOR EXAMPLE, IF YOU THINK (a) IS THE ANSWER TO QUESTION 1, YOU WOULD WRITE IN YOUR ANSWER BOOK: 1. (a).

1. Assume X1, X2, …, X81 to be 81 independent random samples selected from a population with a mean of 20 (i.e. μ = 20) and standard deviation of 9 (i.e. σ = 9). According to the Central Limit Theorem, which one of the following best describes the sampling distribution of the sample means, i.e. ¯X.
(a) ¯X ∼ Normal(mean = 20, variance = 9)
(b) ¯X ∼ Normal(mean = 20, variance = 3)
(c) ¯X ∼ Normal(mean = 20, variance = 1)
(d) ¯X ∼ Normal(mean = 0, variance = 1)
(e) ¯X ∼ t(81)
(f) ¯X ∼ t(80)

2. A random variable X is measured for a representative sample of individuals from a population. If the values of X are known to vary by a standard deviation of σ = 15, what is the minimum sample size required to ensure a 95% confidence interval for the true mean μ has a margin of the error of no more than 5?
(a) 6
(b) 11
(c) 35
(d) 30
(e) 226
(f) 100

3. A random sample of 20 mobile phone users spent, on average, 110 minutes on their phone
with a sample standard deviation of 25 minutes. Which of the following is a 90% confidence
interval for the average time spent on a mobile phone in the population of interest, if we
assume time spent on a mobile phone follows a normal distribution in the population?
(a) 110 ± 9.22
(b) 110 ± 10.96
(c) 110 ± 14.42
(d) 110 ± 11.18

4. A random sample of 100 University of Galway undergraduate students were asked how much they paid each week on rent. The sample mean rent is 596.76. The output below shows the corresponding 95%/95% tolerance interval. Which one of the following interpretations of this tolerance interval is correct?

library(tolerance)
normtol.int(x=rent.df, alpha = 0.05, P = 0.95, side = 2)

alpha P x.bar 2-sided. lower 2-sided.upper
1 0.05 0.95 596.7585 547.4187 646.0982

(a) We are 95% confident that the population average rent paid by University of Galway undergraduate students is between 547.4187 and 646.0982.
(b) We are 95% confident that the average rent paid by University of Galway undergraduate students in this sample is between 547.4187 and 646.0982.
(c) We are 95% confident that the population average rent paid by 95% of University of Galway undergraduate students are between 547.4187 and 646.0982.
(d) We are 95% confident that the sample average rent paid is between 547.4187 and 646.0982.

5. A large supermarket chain wants to estimate the proportion of their Irish customer base that use its ‘scan as you shop’ service. A survey completed by a random sample of 100 shoppers \across a selection of their stores showed that 58 of them used the scanner while shopping, i.e. a sample proportion of 0.58. Which of the following is a 95% confidence interval for the proportion of customers in the population that use their ‘scan as you shop’ service?
(a) 58 ± 0.814
(b) 0.58 ± 0.814
(c) 58 ± 0.967
(d) 0.58 ± 0.0967
(e) 58 ± 0.1273
(f) 0.58 ± 0.1273

6. A researcher conducted a large sample two-sided test of the null hypothesis that H0 : μ = 50. She reports a p-value of p = 0.52. Which one of the following is correct?
(a) The 95% confidence interval for μ will not contain 50
(b) The 95% confidence interval for μ will contain 50
(c) The null hypothesis is rejected at α = 0.05.
(d) The null hypothesis is rejected at α = 0.01.

7. In a hypothesis test with null hypothesis H0, which one of the following statements is true?
(a) If there is insufficient evidence to reject H0, then H0 is true.
(b) If the sample provides evidence to reject H0 when H0 is actually true, this is a Type I
error.
(c) The power of the test is the probability of rejecting H0 when H0 is actually true.
(d) The p-value is the probability of the null hypothesis being true given the data observed.
(e) The significance level is the probability of making a Type II error.

8. In order to test the hypotheses H0: μ = 100 versus Ha: μ < 100, a sample of n = 50, individuals was selected at random from a population. The p-value from the corresponding one-sided test was p = 0.03. Which one of the following is true?
(a) There is evidence to reject H0 : μ = 100 at α = 0.05.
(b) There is evidence to reject H0 : μ = 100 at α = 0.01.
(c) There is insufficient evidence to reject H0 : μ = 100 at α = 0.05.
(d) There is insufficient evidence to reject H0 : μ = 100 at α = 0.01.

9. A researcher has calculated a 95% confidence interval for a population mean based on a random sample of size 20. They are concerned that the normality assumption may not be justified as a boxplot of the variable of interest suggests the data may be right skewed. Which of the following approaches would you not recommend?
(a) Generate a 95% confidence interval for the population median and compare the results.
(b) No action is needed as the sample was chosen at random and therefore the normality assumption is less relevant in this scenario.
(c) Use a suitable transformation of the variable of interest and compare the results.
(d) Generate a 99% confidence interval instead.

10. In order to devise a method to estimate daily mobile phone usage in a particular population of interest, a model is being formulated to predict daily mobile phone usage based on the age of the customer. The phone usage (minutes) and age (years) of a random sample of 20 customers were observed and a simple linear regression, Yi = β0 + β1Xi + i, fitted to the sample data giving the following output.
fit <- lm(minutes ~ age)
summary(fit)

Call:
lm(formula = minutes ~ age)

Residuals:
Min 1Q Median 3Q Max
-70.473 -31.600 -4.342 34.331 94.630

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 329.431 72.141 4.566 0.000239 ***
age 11.782 4.349 2.709 0.014372 *
—Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1

Residual standard error: 47.54 on 18 degrees of freedom
Multiple R-squared: 0.2896, Adjusted R-squared: 0.2502
F-statistic: 7.339 on 1 and 18 DF, p-value: 0.01437

Select all of the following statements that are true.

(a) The coefficient of the variable age, indicates the phone usage increases by 11.782 minutes on average for each 1-year increase in age.
(b) The estimated constant, 329.431, implies the phone usage increases by 329.431 minutes on average for each 1 year increase in age.
(c) s = 47.54 is an estimate of σ. The smaller the value of s, the better the fit of the model to the data.
(d) The p-value 0.014372 indicates the age of the customer is not a good predictor of phone usage in the population.
(e) The p-value 0.014372 indicates the age of the customer is a good predictor of phone usage in the population.
(f) Using this model, the phone usage for an individual aged 20 is estimated to be 565.07 minutes